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Comparative case study research.

• Lesley Bartlett Lesley Bartlett University of Wisconsin–Madison
• , and  Frances Vavrus Frances Vavrus University of Minnesota
• https://doi.org/10.1093/acrefore/9780190264093.013.343
• Published online: 26 March 2019

Case studies in the field of education often eschew comparison. However, when scholars forego comparison, they are missing an important opportunity to bolster case studies’ theoretical generalizability. Scholars must examine how disparate epistemologies lead to distinct kinds of qualitative research and different notions of comparison. Expanded notions of comparison include not only the usual logic of contrast or juxtaposition but also a logic of tracing, in order to embrace approaches to comparison that are coherent with critical, constructivist, and interpretive qualitative traditions. Finally, comparative case study researchers consider three axes of comparison : the vertical, which pays attention across levels or scales, from the local through the regional, state, federal, and global; the horizontal, which examines how similar phenomena or policies unfold in distinct locations that are socially produced; and the transversal, which compares over time.

• comparative case studies
• case study research
• comparative case study approach
• epistemology

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Rethinking case study research: A comparative approach

• Organizational Leadership, Policy and Development

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Comparative case studies are an effective qualitative tool for researching the impact of policy and practice in various fields of social research, including education. Developed in response to the inadequacy of traditional case study approaches, comparative case studies are highly effective because of their ability to synthesize information across time and space. In Rethinking Case Study Research: A Comparative Approach, the authors describe, explain, and illustrate the horizontal, vertical, and transversal axes of comparative case studies in order to help readers develop their own comparative case study research designs. In six concise chapters, two experts employ geographically distinct case studies-from Tanzania to Guatemala to the U.S.-to show how this innovative approach applies to the operation of policy and practice across multiple social fields. With examples and activities from anthropology, development studies, and policy studies, this volume is written for researchers, especially graduate students, in the fields of education and the interpretive social sciences.

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• Comparative Case Study Keyphrases 100%
• Case Study Research Keyphrases 100%
• Case Study Social Sciences 100%
• Research Design Economics, Econometrics and Finance 100%
• Anthropology Keyphrases 25%
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T1 - Rethinking case study research

T2 - A comparative approach

AU - Bartlett, Lesley

AU - Vavrus, Frances

N1 - Publisher Copyright: © 2017 Taylor & Francis. All rights reserved.

PY - 2016/11/10

Y1 - 2016/11/10

N2 - Comparative case studies are an effective qualitative tool for researching the impact of policy and practice in various fields of social research, including education. Developed in response to the inadequacy of traditional case study approaches, comparative case studies are highly effective because of their ability to synthesize information across time and space. In Rethinking Case Study Research: A Comparative Approach, the authors describe, explain, and illustrate the horizontal, vertical, and transversal axes of comparative case studies in order to help readers develop their own comparative case study research designs. In six concise chapters, two experts employ geographically distinct case studies-from Tanzania to Guatemala to the U.S.-to show how this innovative approach applies to the operation of policy and practice across multiple social fields. With examples and activities from anthropology, development studies, and policy studies, this volume is written for researchers, especially graduate students, in the fields of education and the interpretive social sciences.

AB - Comparative case studies are an effective qualitative tool for researching the impact of policy and practice in various fields of social research, including education. Developed in response to the inadequacy of traditional case study approaches, comparative case studies are highly effective because of their ability to synthesize information across time and space. In Rethinking Case Study Research: A Comparative Approach, the authors describe, explain, and illustrate the horizontal, vertical, and transversal axes of comparative case studies in order to help readers develop their own comparative case study research designs. In six concise chapters, two experts employ geographically distinct case studies-from Tanzania to Guatemala to the U.S.-to show how this innovative approach applies to the operation of policy and practice across multiple social fields. With examples and activities from anthropology, development studies, and policy studies, this volume is written for researchers, especially graduate students, in the fields of education and the interpretive social sciences.

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U2 - 10.4324/9781315674889

DO - 10.4324/9781315674889

AN - SCOPUS:85021003907

SN - 9781138939516

BT - Rethinking case study research

PB - Taylor and Francis Inc.

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Chapter 20 deals with the set of methods used in comparative case study analysis, which focuses on comparing a small or medium number of cases and qualitative data. Structured case study comparisons are a way to leverage theoretical lessons from particular cases and elicit general insights from a population of phenomena that share certain characteristics. The chapter discusses variable-oriented analysis (guided by frameworks), formal concept analysis and qualitative comparative analysis. It goes on to discuss the types of social-ecological systems (SES) problems and research questions commonly addressed by this set of methods, as well as their limitations, resource implications and new emerging research directions. The chapter also includes an in-depth case study showcasing the application of comparative case study analyses, and suggested further readings on these methods.

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Comparative Case Studies: An Innovative Approach

What is a case study and what is it good for? In this article, we argue for a new approach—the comparative case study approach—that attends simultaneously to macro, meso, and micro dimensions of case-based research. The approach engages two logics of comparison: first, the more common compare and contrast; and second, a 'tracing across' sites or scales. As we explicate our approach, we also contrast it to traditional case study research. We contend that new approaches are necessitated by conceptual shifts in the social sciences, specifically in relation to culture, context, space, place, and comparison itself. We propose that comparative case studies should attend to three axes: horizontal, vertical, and transversal comparison. We conclude by arguing that this revision has the potential to strengthen and enhance case study research in Comparative and International Education, clarifying the unique contributions of qualitative research.

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Abstract Finding the balance between adequately describing the uniqueness of the context of studied phenomena and maintaining sufficient common ground for comparability and analytical generalization has widely been recognized as a key challenge in international comparative research. Methodological reflections on how to adequately cover context and comparability have extensively been discussed for quantitative survey or secondary data research. In addition, most recently, promising methodological considerations for qualitative comparative research have been suggested in comparative fields related to higher education. The article's aim is to connect this discussion to comparative higher education research. Thus, the article discusses recent advancements in the methodology of qualitative international comparative research, connects them to older analytical methods that have been used within the field in the 1960s and 1970s, and demonstrates their analytical value based on their application to a qualitative small-N case study on research groups in diverse organizational contexts in three country contexts.

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This is the inaugural volume in the PSCIE (Pittsburgh Studies in Comparative and International Education) Series which expands on the life work of University of Pittsburgh professor Rolland G. Paulston (1929-2006). Recognized as a stalwart in the field of comparative and international education, Paulston's most widely recognized contribution is social cartography. He demonstrated that mapping comparative, international, and development education is no easy task and, depending on the perspective of the mapper, there may be multiple cartographies to chart. This collection of nineteen essays and research studies is a festschrift celebrating and developing Robert Paulston's scholarship in comparative, international, and development education (CIDE). Considering key international education issues, national education systems, and social and educational theories, essays in this volume explore and go beyond Paulston's seminal works in social cartography. Organized into three sec...

Ben Hawbaker , Candace Jones , Brooke Boren , Reut Livne-Tarandach

Qualitative researchers utilize comparative and case-based methods to develop theory through elaboration or abduction. They pursue research in intermediate fields where some but not all relevant constructs are known (Edmonson & McManus, 2007). When cases and comparisons move beyond a few, it threatens researchers with information overload. Qualitative Comparative Analysis (QCA) is a novel method of analysis that is appropriate for larger case or comparative studies and provides a flexible tool for theory elaboration and abduction. Building on recently published exemplars from organizational research, we illuminate three key benefits of QCA: (1) allows researchers to examine cases as wholes, effectively addressing the complexity of action embedded in organizational phenomena; (2) provides indicators of whether results are reliable and valid so qualitative researchers, and others, can assess their findings within a study and across studies; and (3) explores potentially overlooked connections between qualitative and quantitative research.

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This course focuses on how to design and conduct small-n case study and comparative research. Thinking outside of students' areas of interest and specialisms and topics, students will be encouraged to develop the concepts and comparative frameworks that underpin these phenomena. In other words, students will begin to develop their research topics as cases of something. The course covers questions of design and methods of case study research, from single-n to small-n case studies including discussions of process tracing and Mill's methods. The course addresses both the theoretical and methodological discussions that underpin research design as well as the practical questions of how to conduct case study research, including gathering, assessing and using evidence. Examples from the fields of comparative politics, IR, development studies, sociology and European studies will be used throughout the lectures and seminars.

Reut Livne-Tarandach , Candace Jones

Qualitative researchers utilize comparative and case-based methods to develop theory through elaboration or abduction. They pursue research in intermediate fields where some but not all relevant constructs are known (Edmonson & McManus, 2007). When cases and comparisons move beyond a few, it threatens researchers with information overload. Qualitative Comparative Analysis (QCA) is a novel method of analysis that is appropriate for larger case and comparative studies and provides a flexible tool for theory elaboration and abduction. Building on recently published exemplars from organizational research, we illuminate three key benefits of QCA: (1) allows researchers to examine cases as wholes, effectively addressing the complexity of action embedded in organizational phenomena; (2) provides indicators of whether results are reliable and valid so qualitative researchers, and others, can assess their findings within a study and across studies; and (3) explores potentially overlooked connections between qualitative and quantitative research.

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Case study methodology has long been a contested terrain in social sciences research which is characterized by varying, sometimes opposing, approaches espoused by many research methodologists. Despite being one of the most frequently used qualitative research methodologies in educational research, the methodologists do not have a full consensus on the design and implementation of case study, which hampers its full evolution. Focusing on the landmark works of three prominent methodologists, namely Robert Yin, Sharan Merriam, Robert Stake, I attempt to scrutinize the areas where their perspectives diverge, converge and complement one another in varying dimensions of case study research. I aim to help the emerging researchers in the field of education familiarize themselves with the diverse views regarding case study that lead to a vast array of techniques and strategies, out of which they can come up with a combined perspective which best serves their research purpose.

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Rethinking Case Study Research A Comparative Approach

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Comparative case studies are an effective qualitative tool for researching the impact of policy and practice in various fields of social research, including education. Developed in response to the inadequacy of traditional case study approaches, comparative case studies are highly effective because of their ability to synthesize information across time and space. In Rethinking Case Study Research: A Comparative Approach , the authors describe, explain, and illustrate the horizontal, vertical, and transversal axes of comparative case studies in order to help readers develop their own comparative case study research designs. In six concise chapters, two experts employ geographically distinct case studies—from Tanzania to Guatemala to the U.S.—to show how this innovative approach applies to the operation of policy and practice across multiple social fields. With examples and activities from anthropology, development studies, and policy studies, this volume is written for researchers, especially graduate students, in the fields of education and the interpretive social sciences.

Table of Contents

Lesley Bartlett is Professor in the Department of Educational Policy Studies and a faculty affiliate with the Department of Anthropology and the Department of Curriculum and Instruction at the University of Wisconsin-Madison, USA. Frances Vavrus is Professor in the Department of Organizational Leadership, Policy, and Development and a faculty affiliate at the Interdisciplinary Center for the Study of Global Change at the University of Minnesota, USA.

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Kolmogorov-Arnold Networks in Low-Data Regimes: A Comparative Study with Multilayer Perceptrons

Multilayer Perceptrons (MLPs) have long been a cornerstone in deep learning, known for their capacity to model complex relationships. Recently, Kolmogorov-Arnold Networks (KANs) have emerged as a compelling alternative, utilizing highly flexible learnable activation functions directly on network edges, a departure from the neuron-centric approach of MLPs. However, KANs significantly increase the number of learnable parameters, raising concerns about their effectiveness in data-scarce environments. This paper presents a comprehensive comparative study of MLPs and KANs from both algorithmic and experimental perspectives, with a focus on low-data regimes. We introduce an effective technique for designing MLPs with unique, parameterized activation functions for each neuron, enabling a more balanced comparison with KANs. Using empirical evaluations on simulated data and two real-world data sets from medicine and engineering, we explore the trade-offs between model complexity and accuracy, with particular attention to the role of network depth. Our findings show that MLPs with individualized activation functions achieve significantly higher predictive accuracy with only a modest increase in parameters, especially when the sample size is limited to around one hundred. For example, in a three-class classification problem within additive manufacturing, MLPs achieve a median accuracy of 0.91, significantly outperforming KANs, which only reach a median accuracy of 0.53 with default hyperparameters. These results offer valuable insights into the impact of activation function selection in neural networks.

1 Introduction

Multilayer Perceptrons (MLPs) are essential to modern deep learning due to their ability to model intricate nonlinear relationships [ 1 ] . MLPs consist of interconnected layers of neurons, forming a network where information flows from input to output. The connections between neurons in adjacent layers are represented by edges, each associated with a weight that determines the strength of the signal passed between them. Within each hidden layer, every neuron performs two key tasks. The first one is to calculate a weighted sum of its inputs, aggregating the signals received from the neurons in the previous layer. The second task involves applying a nonlinear activation function to this weighted sum. Without the nonlinear transformations provided by activation functions, neural networks would be limited to simple linear relationships, severely hindering their ability to extract complex patterns [ 2 ] .

Traditionally, MLPs relied on fixed activation functions such as the Rectified Linear Unit (ReLU) [ 3 ] or the hyperbolic tangent (tanh). Training in this paradigm focuses mainly on adjusting edge weights to reduce error, keeping the activation functions constant. Recently, there has been a growing interest in parameterized activation functions, which offer greater control during the learning process [ 4 , 5 ] . These functions contain adjustable or learnable parameters, allowing the network to fine-tune its nonlinearities as it trains. This enhanced flexibility can improve performance by tailoring the model’s internal transformations to the unique characteristics of the data.

In the pursuit of adaptive activation functions, significant research has focused on parameterizing functions that build upon ReLU, allowing for more nuanced handling of negative inputs. For instance, Leaky ReLU introduces a slight linear slope for negative values [ 6 ] , while the Exponential Linear Unit (ELU) employs an exponential function in this region [ 7 ] . Additionally, the Sigmoid Linear Unit (SiLU), also known as the swish function [ 8 , 9 , 10 ] , has garnered attention for its smooth, non-step-like behavior and its ability to approximate both linear and ReLU characteristics depending on its parameter. Several empirical studies have demonstrated the superior performance of learnable activation functions in large-scale problems involving tens of thousands of samples or more [ 11 ] .

Recently, Kolmogorov-Arnold Networks (KANs) have been introduced as a novel neural network architecture [ 12 ] . Unlike traditional MLPs, KANs place learnable activation functions on the connections or edges between neurons, rather than within the neurons themselves. This design is rooted in the mathematical principle that multivariate functions can be decomposed into simpler univariate ones using sums [ 13 ] . Figure 1 illustrates the structural contrast between MLPs and KANs. In MLPs, each neuron performs both summation of weighted inputs and applies a nonlinear activation function, while in KANs, nonlinearity is introduced through the edges of the network themselves.

The recent KAN research [ 12 ] introduces the concept of a KAN layer, effectively stacking the neurons depicted in Figure 1 (b), and provides an implementation within a widely-used deep learning framework. This allows for the creation of arbitrarily deep networks through automatic differentiation. In practice, KANs often parameterize their edge activation functions using a combination of a SiLU function and a spline function. Training a KAN then primarily focuses on learning the optimal coefficients for these local B-spline basis functions.

While KANs are designed to utilize more expressive activation functions than standard MLPs with ReLU-like learnable activations, this expressivity comes at the cost of increased parameter count. This raises concerns about their performance in data-sparse scientific and engineering domains, where small sample sizes can hinder effective training. In many fields, such as medicine and engineering, data collection is often limited to a few tens or hundreds of samples due to high costs and time-consuming procedures [ 14 , 15 ] .

Hence, the primary goal of this paper is to provide a comprehensive comparison between MLPs and KANs in low-data scenarios with a few hundred samples. Our key contributions are listed in the following.

Enabling Fair Comparison: Most deep learning libraries implement MLPs with the same activation function applied to all neurons within a layer. Even when using learnable activation functions, neurons in the same hidden layer typically share the same set of parameters. To ensure a fair comparison with KANs, we present a straightforward yet effective technique for designing and implementing MLPs where each neuron in a hidden layer has its own distinct, parameterized activation function. This approach can be applied to networks of any depth or width. Furthermore, we implement a parameterized version of the SiLU activation function to ensure that the activation functions used in MLPs are comparable to those used in KANs.

Mathematical Connections: We present a mathematical analysis that elucidates the relationship between MLPs and KANs, showing that KANs can essentially be considered MLPs with activation functions possessing greater flexibility. This analysis underscores the importance of explicitly comparing the number of learnable parameters for both MLPs and KANs in our empirical study.

Empirical Evaluation: We conduct experiments on a simulated data set and two real-world classification problems (cancer detection and 3D printer type prediction) to investigate the trade-offs between model complexity (parameter count) and accuracy in data-limited settings. As a key feature of recently introduced KANs is their ability to stack multiple KAN layers, we specifically examine the impact of network depth on testing accuracy across various data splits. Additionally, our experiments reveal that the piecewise polynomial order of splines significantly influences the performance of KANs.

The remainder of this paper is organized as follows. In Section 2 , we provide a concise mathematical introduction to MLPs, detail our modification to design MLPs with individual learnable activation functions for each neuron in hidden layers, and explain the underlying transformation in KANs, along with connections to MLPs with trainable activation functions. In Section 3 , we report experiments on simulated data sets to understand the impact of data size on the predictive accuracy of MLPs and KANs. In Section 4 , we present numerical experiments using real-world data representing complex problems in medicine and engineering. Finally, we conclude this paper with remarks in Section 5 .

2 Foundations

𝐿 1 l=1,\ldots,L+1 italic_l = 1 , … , italic_L + 1 . The predicted output f ⁢ ( x ) 𝑓 𝑥 f(x) italic_f ( italic_x ) is then defined from the input x 𝑥 x italic_x according to the following equations:

In contrast, we have more flexibility when choosing activation functions for the L 𝐿 L italic_L hidden layers, as they produce latent representations. Note that in the equation above, the activation function is applied element-wise. Consequently, the standard implementation of dense layers in popular deep learning libraries like TensorFlow/Keras applies the same activation function to all neurons within a given layer. In other words, every neuron in a layer employs the same nonlinear function to transmit information to the subsequent layer.

1 superscript 𝑒 𝛽 𝑧 \text{SiLU}(z;\beta)=z/(1+e^{-\beta z}) SiLU ( italic_z ; italic_β ) = italic_z / ( 1 + italic_e start_POSTSUPERSCRIPT - italic_β italic_z end_POSTSUPERSCRIPT ) is its smooth, non-step-like behavior. Its behavior can transition between linear (when β = 0 𝛽 0 \beta=0 italic_β = 0 ) and ReLU-like (when β 𝛽 \beta italic_β is sufficiently large). In many cases, such choices, like the value of β 𝛽 \beta italic_β , are predefined hyperparameters and not optimized during training.

2.1 MLPs with Adaptive Activation Functions

One approach to enhancing the performance and adaptability of MLPs is to treat the parameters within activation functions as trainable during the learning process. This involves incorporating these parameters into the computation graph, allowing the calculation of gradients of the loss function with respect to them. As a result, these activation function parameters, such as the optimal value of β 𝛽 \beta italic_β for SiLU, can be learned alongside the network weights.

While several empirical studies [ 11 , 20 , 21 ] have demonstrated the advantages of adaptive activation functions over fixed ones in MLPs, particularly in large-scale data settings like computer vision, it is common to assume that all neurons within a layer share the same activation function parameters. In other words, in Equation ( 1 ), g ( l ) superscript 𝑔 𝑙 g^{(l)} italic_g start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT is identical for all N l subscript 𝑁 𝑙 N_{l} italic_N start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT neurons in the l 𝑙 l italic_l -th layer. However, for a fair comparison with KANs, where each connection has its own unique activation function, it is crucial to allow each activation function in MLPs to operate independently, increasing their expressivity.

To achieve this goal, we utilize the built-in concatenation layer available in deep learning libraries, enabling us to seamlessly merge the outputs of neurons with distinct activation functions. Let W i ( l ) superscript subscript 𝑊 𝑖 𝑙 W_{i}^{(l)} italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT denote the i 𝑖 i italic_i -th row of the weight matrix W ( l ) superscript 𝑊 𝑙 W^{(l)} italic_W start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT , and let b i ( l ) superscript subscript 𝑏 𝑖 𝑙 b_{i}^{(l)} italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT represent the i 𝑖 i italic_i -th element of b ( l ) superscript 𝑏 𝑙 b^{(l)} italic_b start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT . Then, each neuron in the l 𝑙 l italic_l -th layer computes its output as follows:

where g i ( l ) superscript subscript 𝑔 𝑖 𝑙 g_{i}^{(l)} italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT is the activation function specific to the i 𝑖 i italic_i -th neuron, with its own learnable parameter β i subscript 𝛽 𝑖 \beta_{i} italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT . The outputs from all N l subscript 𝑁 𝑙 N_{l} italic_N start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT neurons are then concatenated to form the final output of the l 𝑙 l italic_l -th hidden layer:

This approach offers significant flexibility, allowing us to design and implement MLPs of any depth and width. Each neuron has its own unique activation function, and the entire process leverages standard automatic differentiation within existing deep learning libraries.

2.2 Kolmogorov-Arnold Networks (KANs)

KANs depart from the traditional MLP architecture by placing learnable functions directly on the network’s edges (connections between neurons), rather than within the neurons themselves. Consequently, the primary role of each neuron is to simply sum its incoming signals, without applying any additional nonlinearities. This design aims to integrate the nonlinear transformation directly into the weighting mechanism of the edges, rather than separating linear weighting and nonlinear activation as in conventional MLPs.

Consequently, the transformation performed by each layer remains expressible as a matrix-vector multiplication. However, a crucial aspect of designing KANs lies in the selection of activation functions. For simplicity, let us omit all superscripts and subscripts in the following discussion. It has been proposed that each activation function in Equation ( 4 ) can be represented as the weighted sum of SiLU and a spline function:

1 superscript 𝑒 𝑥 \text{SiLU}(x)=x/(1+e^{-x}) SiLU ( italic_x ) = italic_x / ( 1 + italic_e start_POSTSUPERSCRIPT - italic_x end_POSTSUPERSCRIPT ) represents the Sigmoid Linear Unit function with β = 1 𝛽 1 \beta=1 italic_β = 1 and spline ⁢ ( x ) = ∑ i c i ⁢ B i ⁢ ( x ) spline 𝑥 subscript 𝑖 subscript 𝑐 𝑖 subscript 𝐵 𝑖 𝑥 \text{spline}(x)=\sum_{i}c_{i}B_{i}(x) spline ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) is a linear combination of B-splines [ 22 , 23 ] . Thus, the training process involves learning the optimal values of c i subscript 𝑐 𝑖 c_{i} italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , w b subscript 𝑤 𝑏 w_{b} italic_w start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT , and w s subscript 𝑤 𝑠 w_{s} italic_w start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT . This approach enables high expressivity by utilizing rich activation functions that go beyond the popular ReLU function typically used in MLPs. Moreover, both the degree of each spline (spline order) and the number of splines used for each function are hyperparameters of the KAN architecture.

2.3 Bridging MLPs and KANs

While KANs clearly utilize activation functions with greater flexibility than standard MLPs employing functions like SiLU, it is worth questioning whether the KAN architecture is truly novel. Comparing Equations ( 1 ) and ( 4 ) reveals that layer transformations in both architectures can be expressed as matrix-vector multiplications. However, a key distinction lies in the order of operations. MLPs first compute a weighted sum of inputs from the previous layer, followed by a nonlinear activation function. In contrast, KANs apply nonlinear transformations to the inputs first, then perform a weighted sum.

To explain this, we rewrite Equation ( 5 ):

Therefore, the activation function used on KAN edges can be viewed as a nonlinear transformation followed by a weighted sum. This is similar to MLPs, except for the order of these operations. From a practical standpoint, the critical question is whether this difference in ordering, along with the increased number of trainable parameters in KANs, leads to higher accuracy levels compared to MLPs with trainable activation functions, particularly in low-data regimes. The remainder of this paper will focus on investigating this question.

3 Performance Evaluation: Synthetic Data

In this section, we commence our comparative study between MLPs and KANs using a two-dimensional simulated data set. This data set encompasses two classes, each consisting of two clusters. Our objective is to evaluate their performance on a separable yet relatively complex data set. Figure 2 illustrates the two data sets central to our analysis: data set A (1,000 samples) and data set B (100 samples). This enables us to directly understand the impact of data size on the performance of KANs, which possess higher degrees of freedom due to the choice of activation functions; see Equation ( 5 ). For consistency, the default values of grid size 3 and spline order 3 are employed for KANs in this paper, with the exception of Section 4.3 where we investigate the impact of hyperparameter adjustments.

Furthermore, to ensure a fair comparison, we utilize the parameterized SiLU activation for the MLP architecture with the learnable parameter β 𝛽 \beta italic_β . As previously discussed in Section 2.1 , each neuron in this MLP architecture possesses its own set of trainable parameters, achieved through the concatenation process. For model fitting, we consistently use 20 epochs and a fixed learning rate of 0.05 across all experiments in this paper.

Our comparative analysis focuses on varying network depth L 𝐿 L italic_L while maintaining a fixed width (number of neurons per hidden layer) to ensure a compact network for evaluating performance in low-data regimes and align with the KAN paper’s focus on the KAN layer’s stacking potential for deeper models. All experiments use a width of 2 and consider depths of 1, 2, and 3, employing a 70-30 train-test split and measuring accuracy on the testing set, repeated 25 times to account for the impact of data splits and weight initializations.

For our visualizations, we select violin plots, which provide a richer understanding of the data distribution compared to traditional box plots. The width of each violin at a particular value reflects the concentration of data points around that value, while the vertical axis represents the probability density. Additionally, a tick mark within each violin pinpoints the median of the associated evaluation metric.

As shown in Figure 3 (a), both models perform well on data set A due to its separability, resulting in classification accuracies near 1. However, it is noteworthy that both models experience a slight decrease in accuracy as depth increases. Interestingly, this decrease is slightly more pronounced in KANs. For instance, KANs with a depth of 3 reach a minimum accuracy of 0.973, the lowest observed accuracy level. Furthermore, the median accuracy for KANs with depth 3 is 0.990, compared to approximately 0.993 for all MLP depths.

Furthermore, Figure 3 (b) reveals a larger performance gap between MLPs and KANs on the smaller data set (data set B). While MLPs maintain accuracy levels comparable to the previous data set (e.g., median values near 1), the performance of KANs noticeably degrades in this small-scale scenario. For instance, KANs with depth 3 have a median accuracy of 0.983 and a minimum accuracy of 0.933, a significant drop compared to the minimum of 0.973 observed for data set A. When comparing the distribution of results across all depths, MLPs show a clear advantage over KANs.

To gain deeper insights into the behavior of MLPs and KANs in relation to depth, we present the total number of learnable parameters in Figure 4 . This reveals striking differences between the two models due to their degrees of freedom. KANs, using more complex spline functions compared to the parameterized SiLU in MLPs, have an order of magnitude more parameters to learn. For example, at depth 3, MLPs have 27 learnable parameters while KANs have 192. This suggests that even with distinct activation functions per neuron, the parameterization used by MLPs provides sufficient flexibility to effectively classify this data set with its two clusters per class. Hence, this analysis highlights that MLPs have a distinct advantage over KANs when dealing with limited labeled data.

4 Real-World Case Studies

4.1 cancer detection.

In this case study, we use the breast cancer wisconsin data set from scikit-learn, which is a well-known data set in machine learning and medical research. It consists of diagnostic information about breast cancer, including input features derived from digitized images of fine needle aspirates (FNAs) of breast masses. The data set contains 569 samples and each sample is described by 30 real-valued input features, thus D = 30 𝐷 30 D=30 italic_D = 30 . There are 212 malignant and 357 benign samples in the data set, providing a somewhat balanced view of both malignant and benign cases, with the goal of training machine learning models to classify the diagnosis based on the input features.

As detailed in the previous section, we maintain a constant network width of 2 while varying the depth L 𝐿 L italic_L (1, 2, and 3). We also ensure consistency by using the same number of epochs and learning rate values. To assess classification accuracies, we conduct 25 independent experiments, each with its own train-test split and weight initialization. Our goal is to compare the performance of MLPs and KANs across these 25 repetitions using violin plots.

Figure 5 demonstrates that MLPs significantly outperform KANs in this cancer detection problem. The maximum and minimum accuracy levels for MLPs across all three depths are approximately 0.99 and 0.94, respectively. Therefore, our implementation of MLPs with individually trainable parameters in their SiLU activation functions aligns with existing neural network implementations on this popular data set, which have achieved high accuracies approaching 0.99 [ 24 ] .

In contrast, the maximum and minimum accuracy values for KANs are around 0.98 and 0.88, respectively. This wider range of accuracy values, approximately twice that of MLPs, suggests lower overall accuracy and potentially less reliability in their predictions. Similar results are observed when comparing median accuracy levels. For instance, at depth 3, the median accuracy of MLPs is 0.976, while KANs have a substantially lower median value of 0.947. Importantly, MLPs’ performance remains fairly consistent across the three depths, suggesting that hyperparameter tuning may be more straightforward compared to KANs.

To further analyze the impact of activation function choice on model complexity, we present the total number of learnable parameters for MLPs and KANs in Figure 6 , particularly considering the 30-dimensional input. The figure reveals a stark contrast: KANs generally possess an order of magnitude more parameters than MLPs. Despite this increased complexity, KANs underperform in this real-world cancer detection task. This suggests that MLPs, even with individual parameterized activation functions, achieve sufficient complexity for accurate classification using fewer parameters. In this case, the simpler MLP architecture appears to be more effective at learning from the 30 input features, highlighting the potential advantages of parameter efficiency in low-data regimes.

4.2 3D Printer Type Prediction

In fused filament fabrication (FFF), the mechanical properties of printed parts are influenced not only by printing parameters but also by the specific 3D printer used. Variations in hardware and firmware across printer models lead to differences in material deposition, movement precision, and temperature control, impacting factors like interlayer adhesion and ultimately the final part’s strength and surface quality [ 25 ] .

In this section, we evaluate classification models based on MLPs and KANs with varying depths for identifying the 3D printer used to manufacture a given part. Using a data set [ 26 ] comprising tensile properties of parts printed on three different printers (MakerBot Replicator 2X, Ultimaker 3, and Zortrax M200) with varying printing parameters, our models aim to predict the printer type based on 7 input features: tensile strength, elastic modulus, elongation at break, extrusion temperature, layer height, print bed temperature, and print speed. This approach seeks to capture the subtle relationships between printing process, part properties, and the specific printer used, which can be viewed as a three-class classification problem with D = 7 𝐷 7 D=7 italic_D = 7 input features and 104 samples. Thus, this section aims to evaluate the performance of MLPs and KANs in scenarios with severely limited data, a common challenge in many experimental settings.

Figure 7 presents classification accuracy results on this data set across 25 repetitions, each utilizing a 70-30 train-test split ratio, to capture the effects of weight initialization and other stochastic factors in training neural network models. Overall, MLPs demonstrate strong performance across all three depth values. For instance, with a depth of 3, the maximum, median, and minimum accuracies are 1, 0.906, and 0.844, respectively. These results highlight the effectiveness of MLPs, especially considering this is a three-class classification problem where a random classifier would yield accuracies closer to 0.333, significantly lower than our observed results.

On the other hand, we observe a more pronounced accuracy drop in KANs compared to the previous cancer detection data set. While KANs can achieve accuracies at or near 1 in some repetitions, there are instances where accuracy falls below 0.333, the baseline for a random classifier. Fortunately, the median values for KANs at depths 1, 2, and 3 are 0.531, 0.406, and 0.406, respectively, all surpassing the baseline. Nevertheless, MLPs consistently outperform KANs in this scenario.

Similar to previous cases, we also report the total number of learnable parameters on the 3D printer type prediction data set in Figure 8 . Again, we observe that MLPs have approximately an order of magnitude fewer parameters to learn, which is significant given the small sample size. Specifically, the total number of learnable parameters for MLPs at depths 1, 2, and 3 are 27, 35, and 43, respectively. These values are below the sample size, which may explain the consistent good performance of MLPs on the test set across different repetitions.

4.3 Dependence of KANs on the Polynomial Order of Activations

In this section, we delve deeper into the performance of KANs on these two real-world data sets to understand how the complexity of the activation functions influences their behavior. Recall that each activation function in KANs is parameterized as a B-spline, and a crucial hyperparameter is the polynomial order of these splines. An order of 1 results in an activation function similar to ReLU, i.e., a piecewise linear function, while higher orders provide increasing degrees of nonlinearity.

The default spline order in the KAN implementation is 3, offering a reasonable balance of nonlinearity. To gain further insight into the impact of this choice in low-data regimes, we consider networks of depth 2 and width 2, but we vary the spline order from 1 to 5.

Figure 9 (a) reveals an interesting trend: in most cases, the accuracy of KANs on the cancer detection data set decreases as the spline order increases. The highest median accuracy is achieved at order 2 (0.959), while the lowest is at order 5 (0.929). However, MLPs with parameterized swish activations per neuron still outperform the best KAN configuration in this experiment. The median accuracy for MLPs with depth 2 reaches 0.965, while maintaining an order of magnitude fewer learnable parameters. This demonstrates that MLPs with individualized parameterized activations can achieve higher accuracy with significantly fewer parameters, a crucial advantage in low-data scenarios.

Furthermore, we investigate the impact of spline order on the 3D printer type prediction data set, which has a substantially smaller sample size and involves a trinary (three-class) classification problem instead of binary. In this case, Figure 9 (b) demonstrates that even a spline order of 1 can lead to a noticeable number of instances where accuracy falls below 0.8, the minimum value observed for MLPs in the previous section. While all spline orders except for 4 can achieve high accuracies close to 1, a serious concern is the potential for substantial performance degradation due to data splits and other stochastic factors in each repetition. Moreover, even with a spline order of 1, KANs have a much higher number of learnable parameters compared to MLPs. This suggests that more compact MLPs can achieve superior accuracy levels in this scenario.

5 Conclusions and Future Directions

While Kolmogorov-Arnold Networks (KANs) offer an intriguing alternative to Multilayer Perceptrons (MLPs) by replacing linear weights with highly expressive activation functions, our findings highlight their notable performance degradation in low-data regimes compared to MLPs. As our algorithmic comparison revealed, the primary strength of KANs lies in their choice of activation functions because they implicitly incorporate a linear weighting mechanism similar to MLPs. This underscores the critical role of activation function complexity in neural network performance. While complex activation functions with greater flexibility might seem like the obvious choice, our findings suggest that simpler alternatives such as the SiLU can provide adequate capacity for some practical applications, particularly in scenarios with limited data availability.

Additionally, our research has revealed that individually parameterized neurons within hidden layers can derive advantages from utilizing independent, individualized activation functions. This approach does not compromise accuracy and opens new avenues for enhancing the predictive power of smaller networks on small-scale data sets. Implementing such MLPs with fully adaptive activation functions is straightforward in popular deep learning libraries like TensorFlow/Keras and PyTorch using concatenation layers. It is imperative to include these MLPs in future comparative studies, as existing benchmarks focusing on fixed-shape activation functions, e.g., [ 27 ] , may not provide a fair comparison. This paves the way for exploring novel activation functions that offer controlled nonlinear transformations for analyzing complex data.

Several recent works have explored activation functions beyond splines in KANs, which can also be applied to MLPs with individualized activations. For example, wavelet functions can capture both high-frequency and low-frequency components of input data [ 28 ] . Another promising direction is developing adaptive algorithms for selecting activation functions per neuron in MLPs, considering a predefined space including splines, wavelets, Chebyshev polynomials, and others [ 29 , 30 ] . Such an approach could factor in sample size and data complexity measures, proving especially beneficial in low-data scenarios.

Finally, future comparative studies should also investigate the sensitivity of MLPs and KANs to their hyperparameters. Our experiments demonstrated that MLPs are relatively insensitive to network depth, whereas KAN performance can significantly degrade with increasing depth or spline order. A thorough comparative analysis necessitates a comprehensive examination of other hyperparameters like learning rate, number of epochs, and grid size, to name a few.

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Energy efficiency—case study for households in poland.

1. Introduction

• increasing the precision of measurement by increasing the reading frequency, which is essential mainly for the distribution network operator;
• obtaining information on daily electricity consumption by analyzing the amount of daily electricity consumption in the household;
• obtaining information on the household zone with the highest consumption by analyzing the electricity consumption in a specific zone;
• increasing household savings by analyzing energy receivers’ operating time [ 14 ].
• lower power consumption;
• lower supply voltage;
• high efficiency;
• low energy losses;
• smaller size;
• high durability and shock resistance;
• high luminance value;
• possibility of selecting a light color;
• in some models, the ability to control the lighting and its color is achieved via the bulb’s Wi-Fi module.
• appropriate building architecture (building geometry, location, size of transparent partitions, room layout);
• insulation of building partitions;
• type of ventilation used;
• type and efficiency of the heating system;
• equipment with electrical devices of the highest possible energy class;
• energy management system in the building.
• installations for generating electricity and heat using biomass or biogas obtained in the methane fermentation process;
• hydroelectric power plants (electric heating and power supply for automation and heaters in hybrid installations);
• wind power plants (electric heating and power supply for automation and heaters in hybrid installations);
• production of biofuels or other renewable fuels;
• solar collectors obtain heat directly from solar radiation and photovoltaic cells (electric heating or power supply for automation in other heating devices);
• heat pumps and devices using ambient heat or from the Earth’s interior.

2. Materials and Methods

3.1. energy consumption in households in poland, 3.2. actions aimed at improving energy efficiency in surveyed households, 4. conclusions, author contributions, data availability statement, conflicts of interest.

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Gromada, A.; Trębska, P. Energy Efficiency—Case Study for Households in Poland. Energies 2024 , 17 , 4592. https://doi.org/10.3390/en17184592

Gromada A, Trębska P. Energy Efficiency—Case Study for Households in Poland. Energies . 2024; 17(18):4592. https://doi.org/10.3390/en17184592

Gromada, Arkadiusz, and Paulina Trębska. 2024. "Energy Efficiency—Case Study for Households in Poland" Energies 17, no. 18: 4592. https://doi.org/10.3390/en17184592

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16 Sep 2024  ·  Thomas Fraunholz , Dennis Rall , Tim Köhler , Alfons Schuster , Monika Mayer , Lars Larsen · Edit social preview

In the realm of industrial manufacturing, Artificial Intelligence (AI) is playing an increasing role, from automating existing processes to aiding in the development of new materials and techniques. However, a significant challenge arises in smaller, experimental processes characterized by limited training data availability, questioning the possibility to train AI models in such small data contexts. In this work, we explore the potential of Transfer Learning to address this challenge, specifically investigating the minimum amount of data required to develop a functional AI model. For this purpose, we consider the use case of quality control of Carbon Fiber Reinforced Polymer (CFRP) tape laying in aerospace manufacturing using optical sensors. We investigate the behavior of different open-source computer vision models with a continuous reduction of the training data. Our results show that the amount of data required to successfully train an AI model can be drastically reduced, and the use of smaller models does not necessarily lead to a loss of performance.

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Prioritizing sustainable building design indicators through global SLR and comparative analysis of AHP and SWARA for holistic assessment: a case study of Kabul, Afghanistan

• Research Article
• Published: 11 September 2024
• Volume 9 , article number  139 , ( 2024 )

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• Ahmad Walid Ayoobi 1 , 2 ,
• Gonca Inceoğlu 3 &
• Mehmet Inceoğlu 4  

The growing emphasis on sustainable architecture, addressing environmental, social, and economic concerns, has spurred the development of numerous design strategies and assessment methods. This has resulted in many sustainable building design indicators, posing challenges in their selection and application, particularly in developing countries with limited resources. This study aims to address these gaps by employing a Systematic Literature Review (SLR) to identify all commonly used sustainable building design indicators globally. Subsequently, a comparative analysis of the SWARA and AHP methods was conducted to characterize weighting scores, prioritize and adapt evaluated indicators, and identify suitable methods for their selection and analysis. Furthermore, the study proposed a comprehensive and holistic set of indicators for sustainable building design, targeting architects and policymakers. Within this set of indicators, the study identified five as the most globally applicable and critical for achieving building sustainability. Weighting scores and prioritization of these indicators for Kabul City, largely aligned with common rating system indicators, were as follows: Energy Efficiency (27.92% weighting), Material & Resources (19.57% weighting), Site & Ecology (13.92% weighting), Indoor Environment Quality (7.69% weighting), and Water Efficiency (13.87% weighting). The overall results indicated that both methods AHP and SWARA are highly effective for analyzing and adapting indicators for sustainable design. These findings offer valuable insights and guidance for the analysis of sustainable indicators, fostering the development of holistic design approaches and a rating system. Ultimately, this research contributes to a more sustainable built environment, particularly within the context of Kabul city.

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Acknowledgements

This work was conducted as part of the doctoral dissertation research of the first author, Ahmad Walid Ayoobi.

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Faculty of Construction, Department of Architecture, Kabul Polytechnic University, Kabul, Afghanistan

Ahmad Walid Ayoobi

Graduate School of Sciences, Department of Architecture, Eskisehir Technical University, Eskisehir, Turkey

Faculty of Education, Department of Mathematics and Science Education, Anadolu University, Eskisehir, Turkey

Gonca Inceoğlu

Faculty of Architecture & Design, Department of Architecture, Eskisehir Technical University, Eskisehir, Turkey

Mehmet Inceoğlu

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Contributions

A.W.A. conceived the overall research design. A.W.A. was responsible for defining the scope and methodology of the systematic literature review used to identify and evaluate sustainable building design indicators on a global scale. A.W.A. collaborated on data collection for the study. A.W.A. and G.I. jointly conducted the multi-criteria decision-making (MCDM) analysis to determine the relative importance and ranking of sustainable building indicators. A.W.A. and M.I. collaboratively interpreted the data obtained from the literature review and MCDM analysis. A.W.A. drafted the initial manuscript. All authors (A.W.A., M.I., and G.I.) reviewed and approved the final version of the manuscript.

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Ayoobi, A.W., Inceoğlu, G. & Inceoğlu, M. Prioritizing sustainable building design indicators through global SLR and comparative analysis of AHP and SWARA for holistic assessment: a case study of Kabul, Afghanistan. J Build Rehabil 9 , 139 (2024). https://doi.org/10.1007/s41024-024-00494-4

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Received : 09 June 2024

Revised : 20 August 2024

Accepted : 03 September 2024

Published : 11 September 2024

DOI : https://doi.org/10.1007/s41024-024-00494-4

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